Predicting Item Difficulties from item characteristics

Raven's (1986) progressive matrices are used to measure non-verbal intelligence: the capacity to reason by analogy, form comparisons, and think logically. The Coloured Progressive Matrices (CPM) is the 36- item version of the test used with children ages 5-12.

The complex and visual CPM items are not well understood, so we identified 14 characteristics that we thought might differentiate item difficulty (see Table 1, and Green and Kluever, 1992). Each CPM item was rated on each characteristic. Each characteristic was modelled to have its own rating scale. The 14 characteristics were then treated as agents and the 36 CPM items as objects in a Rasch analysis. The purpose of this analysis was to construct a conceptual difficulty for each CPM item based on the number and complexity of its characteristics.

CPM item characteristic map

Figure 1 shows the measures of characteristics and items. CPM items with high measures exhibited many characteristics. Characteristics with high measures were exhibited by few CPM items. Reversal (background to foreground) and B&W (colors limited to black and white) were the least observed characteristics. Only the simplest CPM items have less than 6 response Options.

Rating scale for characteristics

Figure 2 shows a map of the ratings associated with each characteristic. According to this map, a change in the number of lines and dimensions in the stem indicates less of a change in overall CPM item complexity than a change in number of elements or number of directions in the options.

To see how well these conceptual difficulty measures capture the empirical difficulties of the CPM items, the 36 CPM items were calibrated for 2 samples. The first sample was 457 1st through 5th grade children from rural southern Colorado. The second sample was 268 relatively gifted 4 through 12-year old children at an educational assessment center at the University of Denver. The results for the two samples were reassuringly similar.

After elimination of 4 CPM items as too easy or markedly misfitting, Figure 3 shows the plot of 32 conceptual item difficulties against empirically derived ones for sample 2. The correlation is .66. This implies that, after correcting for measurement error, our initial set of 14 characteristics has already explained about half the variance in item difficulties

Empirical vs. Conceptual Item Difficulties

A commonly used alternative approach to item difficulty decomposition is to regress the empirical item difficulties on the characteristics. But the item characteristics are multicollinear, so regression results are heavily influenced by the CPM-item-dependent variances in the characteristic ratings. If the characteristic rating variances were to change, even slightly (e.g., by omitting a CPM item), a different characteristic would take precedence in the regression analysis. This perplexes the researcher.

The Rasch approach, however, goes beyond multiple regression by clearly ordering the item characteristics according to their complexity. The level of successful prediction of empirical difficulties suggests that continued development of this set of item characteristics will be fruitful. The item maps provide the clarity needed to advance thought about identified and potential item characteristics and the general nature of CPM item difficulty.

Green KE & Kluever RC (1992) Components of item difficulty of Raven's Matrices. Journal of General Psychology, 119, 189-199.

Raven JC, Court JH & Raven J (1986). Coloured Progressive Matrices. London: H. K. Lewis & Co.

Table 1. Raven's Item Characteristics
Characteristic Rating value Label
In Stem of CPM Item:
Orientation 0=vertical, horizontal; 1=other Orientation
Symmetry 0=symmetrical; 1=asymmetrical Symmetry
Progression in pattern 0=no increase; 1=increase Progress-1
Dimensions in pattern 1-3=number Dimns
Lines 0=straight; 1=curved Curved
Distinct types of lines or solids 1-3=number Lines
Black and White 0=color; 1=black and white B&W
In Options of CPM Item:
Distinct options 2-6=number Options
Options contain rotation 0=no; 1=yes Rotation
Options contain reflection 0=no; 1=yes Reflection
Options contain progression 0=no; 1=yes Progress-2
Directions of options: vertical, horizontal, diagonal 1-3=number Dirns
Number of elements in the design 1-3=number Elements
Reversal between foreground and background 0 =no; 1=yes Reversal

Predicting item difficulties from item characteristics. Green KE, Kluever RC, Wright BD. … Rasch Measurement Transactions, 1994, 8:2 p.354

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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